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Volume 14
Issue 6
10.3390/axioms14060444
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This is an early access version, the complete PDF, HTML, and XML versions will be available soon.
Article

Improving Stability and Reducing Vibrations of the Smooth and Discontinuous Oscillator Using a Proportional–Derivative Controller

by
M. N. Abd El-Salam
1,*,
Rageh K. Hussein
2 and
Sabry M. El-Shourbagy
1
1
Basic Sciences Department, Higher Technological Institute, 10th of Ramadan City 44634, Egypt
2
Physics Department, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11623, Saudi Arabia
*
Author to whom correspondence should be addressed.
Axioms 2025, 14(6), 444; https://doi.org/10.3390/axioms14060444
Submission received: 1 May 2025 / Revised: 29 May 2025 / Accepted: 4 June 2025 / Published: 5 June 2025
Versions Notes

Abstract

In this study, the Proportional and Derivative Controller (PD) is presented as a modified control that combines features of the Proportional Controller (P-Controller) and the Derivative Controller (D-Controller) to suppress the vibrations of the Smooth and Discontinuous Oscillator (SD). The investigated model has been derived as a one-degree-of-freedom. The frequency response equation of the controlled system has been obtained using the perturbation technique up to a second approximation. The influence of the P-Controller, the D-Controller, and the PD-Controller on the SD-Oscillator amplitude has been studied by plotting the time histories. The numerical and approximate simulation established that the PD-Controller can inhibit the system vibration. Finally, there is high closeness between the numerical solutions (from time histories) and the approximate solutions (from perturbation analysis).
Keywords: proportional and derivative controller; smooth and discontinuous oscillator; primary resonance; autonomous dynamical system; stability proportional and derivative controller; smooth and discontinuous oscillator; primary resonance; autonomous dynamical system; stability

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MDPI and ACS Style

Abd El-Salam, M.N.; Hussein, R.K.; El-Shourbagy, S.M. Improving Stability and Reducing Vibrations of the Smooth and Discontinuous Oscillator Using a Proportional–Derivative Controller. Axioms 2025, 14, 444. https://doi.org/10.3390/axioms14060444

AMA Style

Abd El-Salam MN, Hussein RK, El-Shourbagy SM. Improving Stability and Reducing Vibrations of the Smooth and Discontinuous Oscillator Using a Proportional–Derivative Controller. Axioms. 2025; 14(6):444. https://doi.org/10.3390/axioms14060444

Chicago/Turabian Style

Abd El-Salam, M. N., Rageh K. Hussein, and Sabry M. El-Shourbagy. 2025. "Improving Stability and Reducing Vibrations of the Smooth and Discontinuous Oscillator Using a Proportional–Derivative Controller" Axioms 14, no. 6: 444. https://doi.org/10.3390/axioms14060444

APA Style

Abd El-Salam, M. N., Hussein, R. K., & El-Shourbagy, S. M. (2025). Improving Stability and Reducing Vibrations of the Smooth and Discontinuous Oscillator Using a Proportional–Derivative Controller. Axioms, 14(6), 444. https://doi.org/10.3390/axioms14060444

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